The fate of integer quantum Hall e?ect (IQHE) at weak magnetic ?eld is studied numerically in the presence of correlated disorders. We ?nd a systematic ?oat-up and merging picture for extended levels on the low-energy side which results in direct transitions from higher-plateau IQHE states to the insulator. Such direct transitions are controlled by a quantum critical point with a universal scaling form of conductance. The phase diagram is in good agreement with recent experiments. The issue of continuum vs. lattice model is also discussed. 73.40.Hm, 71.30.+h, 73.20.Jc

How extended levels evolve with disorders and the magnetic ?eld B is central to our understanding of the IQHE. Earlier on, Khmel’nitzkii [1] and Laughlin [2] had argued that extended levels should continuously ?oat up towards higher energy with reducing B. And the assumption that extended levels never merge has led to a select rule of the global phase diagram [3] for IQHE, in which a direct transition from a higher-plateau (ν > 1) state to the insulator is prohibited. But direct transitions have been observed in many recent experiments [4–6], which have renewed the theoretical interest to reexamine the ?oat-up picture in IQHE systems. Previous numerical studies in the tight-binding model (TBM) with white-noise disorders have indicated [7–10] the existence of direct transitions from higher IQHE plateau states to the insulator. But the detailed analysis [8] has also revealed that the lattice e?ect plays a central role there: such direct transitions ?rst happen near the band center due to the presence of extended levels carrying negative topological Chern number (a peculiar lattice e?ect) which form a high-energy IQHE-insulator boundary and start to “?oat-down” towards the low-energy regime with increasing disorder or reducing B. During the ?oat-down process, the boundary keeps merging with lower extended levels such that the plateaus disappear in a one-by-one fashion. The key issue is if such a ?oat-down picture, due to the lattice e?ect, is the unique explanation for direct transitions or there exists a di?erent kind of direct transition free of the lattice e?ect. Recall that in a continuum model, there does not exist a high IQHE-insulator boundary as the band center is essentially located at in?nite energy. In this case a levitation of extended levels by disorders is generally expected as discussed [11,12] perturbatively. Since it has been generally believed that the experimental situation should be physically described by the continuum model due to the weakness of magnetic ?elds compared to the bandwidth and low density of charge carriers, it becomes especially interesting whether the ?oat-up of extended levels alone can also lead to a direct transition in the low-energy regime. A ?oat-up of the lowest extended level actually has 1

been seen in the numerical calculation [10] based on TBM but its journey has quickly ended by merging into the ?oat-down IQHE-insulator boundary from the band center. In order to study how such a ?oat-up feature near the band edge evolves, one has to somehow “delay” the ?oating-down process of the high IQHE-insulator boundary. Note that the inter-Landau-level-mixing caused by uncorrelated (white-noise) disorders happens more strongly near the band center, which may enhance the tendency for extended levels near the band center to move down. So one can try to “smooth” the lattice e?ect by introducing short-range correlations among disorders. As a result to be shown below, a ?oat-up process will then become a dominant e?ect for those lower extended levels as the ?oat-down of the high IQHE-insulator boundary is signi?cantly slowed, in contrast to the case in the whitenoise limit [7–9] at similar weak magnetic ?elds. In this Letter, we present a systematic ?oating-up and merging pattern revealed for extended levels near the band edge. Speci?cally, the lowest extended level starts to ?oat upward at stronger disorder or weaker B and eventually emerges into the second lowest extended level to form a new IQHE-insulator boundary on the low-energy side, leading to a ν = 2 → 0 direct transition, while the aforementioned upper IQHE-insulator boundary still remains at high energy. And such a lower IQHE-insulator boundary keeps moving up to merge with higher-energy extended levels to result in 3 ? 0, 4 ? 0, ... direct transitions with increasing disorders or reducing B. The phase diagram is in good agreement with the experiments. Furthermore, direct transitions to the insulator at the lower boundary are found to be consistent with a quantum critical point picture, and in particular the conductance as a function of B ? Bc (Bc denotes the critical magnetic ?eld) is of the universal form for 1 ? 0, 2 ? 0, ... up to 6 ? 0 transitions within the numerical resolutions. The TBM model H = ? <ij> eiaij c+ cj + H.c. + i + i wi ci ci , with the magnetic ?ux per plaquette φ = 2 aij = 2πm/M (m and M are integers). And we de?ne B = m/M . The correlated disorder wi is generated

by wi = W/π j fj e?|Ri ?Rj | /λ0 . Here Ri denotes the spatial position of site i. W and λ0 are the strength and correlation length scale of disorders, respectively. fi is a random number distributing uniformly between (-1,1). To illustrate how extended levels near the band edge evolve with the disorder strength W , the density of states carrying nonzero Chern number (ρext ) as a function of the Landau level ?lling number (nL ) is plotted in Fig. 1 (a) and (b) at B = 1/64, λ0 = 1, and the sample size 32 × 64. The sample-size-independent peaks in the ?gures denote the positions of extended levels while the non-peak part of ρext should scale to zero presumably in the thermodynamic limit [8,13]. In Fig. 1(a), as W is increased from 0.7 to 1.4, extended level positions (marked by diamonds) all start to ?oat up slightly from the ?lling number nL = ν + 0.5 (ν = 0, 1, ...). But the lowest extended level distinctly moves faster and eventually merges into the second lowest extended level to form a new boundary extended level [see W = 1.45 and 1.55 cases in Fig. 1(b)], while the higher extended levels still remain roughly equally spaced (note that in this strong disorder case, the merged peak of ρext becomes less signi?cant but later we will discuss a more reliable way to identify the extended levels). It corresponds to the collapse of the mobility gap separating the lowest two neighboring extended levels and thus the destruction of the ν = 1 IQHE plateau in between. By further increasing W , we see that such a newly merged extended level boundary continuously ?oats up and merges into the third extended level, and so on and so forth. In this way, the IQHE plateaus disappear on the low nL side also in a one by one fashion. We emphasize that higher extended levels originally at nL = ν + 0.5 (ν ≥ 1) never move passing the nL = ν + 1 before merging with the lower boundary [Fig. 1(b)] such that each IQHE plateau does not really ?oat away before its destruction. Such a ?oat-up and merging picture persists into very weak magnetic ?elds (from B = 1/64 to B = 1/2304). The disorder strength Wc at which the merged lowest two extended levels pass nL = 2, resulting in a 2 ? 0 transition, is shown in the inset of Fig. 1(c) as a function of B. We see that Wc monotonically decreases with B and is extrapolated to zero in zero B limit in a fashion of B 1/2 . By following the trace of extended levels in the n ? B plane (n is the on-site electron density) at ?xed W = 1.4 and λ0 = 1, we determine a phase diagram in Fig. 1(c), where the ?lled circles represent the lower IQHEinsulator boundary, while the open circles are the positions of various extended levels between plateaus which merge into the boundary at weaker B. This phase diagram is very similar to the recent experimental phase diagram [Fig. 2 in Ref. [4]] as well as the earlier one obtained in Ref. [5,6]. The Hall conductance calculation con?rms that σxy indeed saturates to νe2 /h in ν-th plateau region while it approches zero on insulating side. Both critical 2

2

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conductances σxxc and σxyc at ν → 0 transition are close to νe2 /2h in accordance with experiments [6]. Apparently the above results critically depend on how reliably one can identify the positions of extended levels using ?nite-size calculations. Let us focus on the merged extended levels as the lower IQHE-insulator boundary shown in Fig. 1(c). By ?xing nL = 2, marked by the arrow C in Fig. 1(c), we calculated the longitudinal conductance σxx with B changing continuously at ?xed W = 1.4. We found a peak in σxx at Bc = 1/70 as a 2 ? 0 transition. In Fig. 2(a), σxx as a function of B with nL = 1, 2, ..., and 6 at W = 1.4 are shown for sample width L = 96 (the stripe sample with Lx = L and Ly ? 106 is considered using transfer matrix method [14]). The L-independent peak positions should correspond to ν = 1 → 0, 2 → 0, ..., and 6 → 0 transitions in Fig. 1(c). Remarkably, all these data can be collapsed onto a universal curve σxx /σc = 2 exp(s)/(1 + exp(2s)), (1)

from L = 32 to L = 160. The correlation length exponent is identi?ed to be x = 4.6 ± 0.5 in the inset of Fig. 3(a), about doubled from x = 2.3 for the ν = 1 → 0 transition which has been similarly determined at nL = 1. The exponents for 3 ? 0, ..., 6 ? 0 transitions seem further increased but are more di?cult to determine with a similar accuracy for larger sample sizes are needed. It is noted that the scaling form (1) is not limited to nL = integer. For example, the data at ν = 2 → 0 transition with nL = 2.2 can be also collapsed onto the same curve. Furthermore, Eq.(1) still holds as we change λ0 from 1 to 2 and 3. Details will be presented elsewhere. Based on (1) and (2), we conclude that the ν = 2 → 0 transition corresponds to a quantum critical point with measure zero in the L → ∞ limit. We note that the same scaling form has been previously obtained [15] for h h the 1?0 transition, where ρxx = exp(?s) e2 and ρxy = e2 leading to (1). Identifying such a simple scaling relation for ν = 1, 2 → 0 as well as higher plateaus to insulator transitions may be the most striking evidence for a single quantum critical point at each transition. The standard scaling method can be also applied to the ν = 2 → 0 transition to independently verify the one parameter scaling law [16]. As shown in Fig. 3(b), by collapsing the same data as σxx (L)/σc = f (ξ/L) by a correlation length ξ, we ?nd ξ ∝ |(B ? Bc )/Bc |?x with x = 4.5 ± 0.5 [the inset of Fig. 3(b)] in agreement with the above result.

We have also checked the case right before the lowest two extended levels merge together. By scanning B at a ?xed nL = 1.8 nearby the scan C shown in Fig. 1(c), σxx exhibits two distinct peaks at Bc1 = 0.0151 and Bc2 = 0.0169 with L = 128 [see the middle inset of Fig. 4(a)]. The main panel of Fig. 4(a) shows the ?nite-size scaling curve of σxx /σc as a function of ξ/L obtained by collapsing the data of di?erent sample sizes at B < Bc1 . The right inset indicates that ξ diverges at Bc1 with an exponent x = 2.4 which is essentially the same as the standard one for the 1 ? 0 transition. Similar ?nite-size scaling curve has been also obtained for the branch at B > Bc2 , corresponding to the 2 ? 1 transition. It is noted that the above analysis resembles the study [17] of the spin unresolved case at strong magnetic ?eld where a small spin-orbit coupling is used to lift the spin degeneracy to create two separated but very close quantum critical points. Similar to the latter case, if one “mistakenly” treats the present case as a single critical point at Bm , a middle point between Bc1 and Bc2 , and proceeds with a ?nite-size scaling analysis, then one gets Fig. 4(b) where the quality of data collapsing becomes markedly worse and in particular ξ shows a saturation trend approaching Bm , contrary to the assumption of a critical point at Bm . Thus, the two separated extended levels with |Bc1 ? Bc2 |/Bm ? 0.11 is not mistakable as a single critical point in our numerical analysis. Finally, we make a remark that even if the 2-0 transition that we observed is actually two transitions with very small |Bc1 ?Bc2 | indistinguishable numerically, the nice scaling properties [(1) and (2)] that we found still indicate that there is a new quantum critical point. In this case, the splitting |Bc1 ? Bc2 | = 0, if it exists, should be caused by a relevant operator of such a new critical point. To summarize, we have identi?ed for the ?rst time a new ?oat-up and merging pattern for extended levels near the band edge at weak B (down to B = 1/2304 where there are 1252 Landau levels between the band edge and center). The corresponding phase diagram with direct transitions is in excellent agreement with the experiments where the essential features can be explained by the narrowing and destruction of each IQHE plateau due to the sequential merging of neighboring extended levels as the mobility gap in between collapses. Acknowledgments - D.N.S. would like to acknowledge helpful discussions with R. N. Bhatt, F.D.M. Haldane, and M. Hilke. D.N.S. and Z.Y.W. are supported by the State of Texas through ARP Grant No. 3652707 and TCSUH. X.G.W. is supported by NSF Grant No. DMR–97–14198 and NSF-MRSEC Grant No. DMR–98– 08941.